July 15-26, 1996
Until comparatively recently, most people tended to view number theory
as the very paradigm of pure mathematics. With the advent of computers,
however, number theory has been finding an increasing number of applications
in practical settings, such as in cryptography, random number generation,
and coding theory. Yet other applications are still emerging, providing
number theorists with some major new areas of opportunity.
This workshop will try to foster the further development of some of these
new applications of number theory, focusing in particular on recent links
with:
- (a) wave phenomena in quantum mechanics (so-called quantum chaos);
- (b) graph theory (especially expander graphs).
Though the central questions of number theory continue to refer to the most
basic of mathematical objects, viz. the integers, the techniques
that are used nowadays to attack these questions frequently come from diverse
areas of mathematics as well as physics and computer science. It is therefore
desirable to bring together number theorists and researchers in these other
areas who have relevant expertise.
This Summer Program will provide a congenial setting where interested workers
from the respective pure and applied groups can come together for two weeks
to listen to a variety of state-of-the-art expository lectures, participate
in discussions concerning current and emerging applications of number theory,
and make new scientific contacts of a cross-disciplinary nature.
During the first week, the program will focus mainly on quantum mechanics.
Whereas classical physics was based primarily on differential equations
-- ordinary as well as partial -- quantum physics has drawn from its very
beginning on mathematics of a more discrete type, in particular group theory
and algebra. The relevance of number theory in such a setting is thus not
entirely unexpected. As the phenomenon of chaos in classical mechanics became
better understood, the spirit of number theory was found to enter physics
even at the classical level, e.g., in the study of critical resonances
and the use of symbolic dynamics. Manifestations of classical chaos are
visible not only in quantum mechanics, but also in a variety of other wave
theories such as optics, electromagnetism, and acoustics.
The distribution of energy levels and eigenfrequencies, as well as their
relation with the classical periodic orbits (through the trace formula),
has led to new methods in spectroscopy which are at least partially based
on number theory. Prime examples are particles moving either in a cavity
or on a Riemann surface of negative curvature, and the resultant description
using zeta functions. Applications of these concepts have recently been
made to small switching elements, quantum computers, polymers, etc.
The second week will see the focus gradually shift over to one of graph
theory. Application-wise, graphs give a very good model for communication
networks, both among people and processors. In many types of analyses, the
optimal network often turns out to be one having properties similar to random
networks. One good example of such a property is "expansion",
which guarantees an absence of "hot spots" in the network. It
turns out that random graphs have a great deal of expansion, but it is a
famous open problem to give explicit constructions of graphs with as much
expansion. A lot of excitement has been generated in recent years with the
discovery of explicit constructions having quite good expansion; these constructions
require number theory and the spectral theory of graphs to prove that they
do, in fact, have the necessary amount of expansion. These developments
also have ties to the spectral theory of differential operators, algebraic
geometry, and representation theory.
The graph-theoretic part of the workshop will concentrate on the use of
number theory to construct graphs having desirable features such as the
good expansion property mentioned above. There are a fair number of ways
number theory can be used to produce a large class of such graphs. To show
that these graphs have desirable properties, one combines the spectral theory
of graphs with number theory to prove that these graphs' eigenvalues are
small, and then checks that any graph with small eigenvalues necessarily
has good behavior. There are a number of relationships between eigenvalues
and graph properties that are not yet well understood, and some of the lectures
will touch on this. In addition, there are a number of new applications
of expanders, including to coding theory, which will be addressed. Finally,
since the properties of number-theoretical graphs are often mimicked by
random graphs, there will be some presentations devoted to random graphs
as well.
|
